Some boundedness properties of solutions to the complex Yang-Mills equations on closed $4$-manifolds

TL;DR

This paper investigates the analytical properties of solutions to the complex Yang-Mills equations on closed four-manifolds, establishing bounds and gap results under specific geometric conditions.

Contribution

It introduces new boundedness properties for solutions when the metric is good and the connection is approximate ASD, leading to gap results for related equations.

Findings

Extra field has a positive lower bound under certain conditions

Establishment of gap results for Yang-Mills and Kapustin-Witten equations

Analytical properties of solutions on closed 4-manifolds

Abstract

In this article, we study the analytical properties of the solutions of the complex Yang-Mills equations on a closed Riemannian four-manifold $X$ with a Riemannian metric $g$. The main result is that if $g$ is $good$ and the connection is an approximate ASD connection, then the extra field has a positive lower {bound}. As an application, we obtain some gap results for Yang-Mills connections and Kapustin-Witten equations.